The Democrats have a somewhat undeserved reputation for being hostile to religion. Some factions that make up the Democrat party absolutely are. Some Democrats in elected office are. And they are really nasty about it. Your Senator Feinstein comes to mind. But on the whole it is not true. The Republicans have been, in the past, regarded as the champions of individual liberties. The freedom of religion being on of the big ones. That was mostly true once but now, not really. Political support is something of a habit I've observed. My father was a red blooded southern Democrat who supported them because they were "for the working man like me" long after that ceased to be true. They guys at the VFW were Reagan converts and have been staunch Republicans ever since because the GOP put America's interests first if that was ever even true. They supported Trump for that same reason. I think, even without the encouragement of religious leaders people who vote on the basis of religious faith will gravitate to the Republicans because that is what they did in the past.
I do agree the strong feelings abortion evokes in people are sincerely felt. But nowhere I've read does the Christian faith charge people to stop the sins of others. In fact we are told to love the sinner and hate the sin. If abortion is a sin it's between the person doing it and God and no one else. We are all countable only for ourselves.